The centre that is not a place

Newtonian mechanics has a theorem worth more than most: whatever a system of particles does internally, one point moves in a straight line at constant speed. Every internal force cancels in pairs, so the centre of mass carries the whole of the momentum and none of the complication, and half of classical mechanics is the business of separating that motion from the rest.

Relativity keeps the theorem and changes the point.

Energy is what does the weighting #

The Newtonian centre weights each position by mass. The relativistic one weights by energy, and the difference is not cosmetic.

In an elastic collision the rest masses are unchanged. If positions were weighted by rest mass, then at the moment of collision the two bodies would exchange velocities without the weights changing, and the weighted point would acquire a kink — it would be a broken line, not a straight one, and the theorem would be gone.

Weight by energy instead and the kink disappears, because the energies are redistributed by exactly the amount that compensates. The figure does not argue this; it computes the outgoing speeds from conservation of energy and momentum, forms the weighted average at two hundred instants, and requires the result to lie on a straight line to a part in $10^{12}$.

The speed of that line is $P/E$ — total momentum over total energy, in units where $c = 1$. That is not the velocity of any particle in the system, and it is not the average of their velocities; it is the velocity of the frame in which the total momentum vanishes, which is the frame the invariant that survives a boost uses to define the system’s mass.

So the theorem survives, with energy in the place mass used to occupy — which is the same substitution that runs through the whole subject, arriving here in the least expected place.

The same theorem at other numbers #

Redrawing at different numbers is worth the space, because a straight line is exactly the shape a figure produces when it has been drawn rather than computed.

In the second arrangement the lighter body carries more momentum than the heavier one, so the centre drifts in the opposite direction from the first case. The outgoing speeds are different, the energy shares are different, and the theorem is untouched: the weighted point runs straight through, and the departure from straightness is again below a part in $10^{12}$.

The generality is worth stating plainly. Nothing in the argument used the collision being elastic. If the two bodies had stuck together, or fragmented into six pieces, or converted half their kinetic energy into a burst of light, the energy-weighted centre would still have run straight — because the only inputs are that the total energy and the total momentum are conserved and that the pieces move at their own velocities. A conservation law about two quantities produces a theorem about a third, and the theorem does not care what happened in between.

That is the same structure as the Newtonian original, where the internal forces cancel in pairs regardless of what they are. What relativity changes is which average is the one that works.

But the point is not the same point #

Here the classical analogy breaks, and it breaks in a way that has no Newtonian shadow at all.

A Galilean observer moving past a system computes the same centre of mass as anybody else, because mass is a Galilean invariant and the weights are the same for everybody. Energy is not an invariant. A body moving fast in one frame is at rest in another, its energy is different in the two, and its share of the system’s total is different.

An energy-weighted average with frame-dependent weights is a frame-dependent point. Two observers looking at the same system will not merely assign different coordinates to one event; they will pick out different events.

There is a second and independent reason for the same conclusion, and it is worth having because it explains why the effect cannot be avoided by any cleverness in the definition. Taking an average over a system’s parts requires taking them all at one time, and two frames disagree about which events are simultaneous. Even if the weights agreed, the slices would not.

What a boost does to a spinning body #

The sharpest form of all this appears for a body with internal angular momentum, and it has a name and a size.

Take a spinning body at rest. Its centre of energy sits at an obvious place. Now boost sideways. The parts of the body moving forwards relative to the boost gain energy and the parts moving backwards lose it, so the energy distribution is no longer symmetric about the geometrical centre, and the weighted centre shifts — perpendicular to both the spin axis and the direction of the boost.

The shift is $S\beta/Mc$, where $S$ is the angular momentum. Sweeping the frame speed from $-c$ to $+c$ sweeps the centre across a disc of radius $S/Mc$, the Møller radius, and every point of that disc is the centre of energy in some inertial frame.

Inside that disc the question has no answer. Not a hard answer, not an answer requiring care — no answer, because “the centre” is defined only relative to a frame and the frames disagree by that much.

This is the same accounting that produces the momentum of something that is not moving, where a static system carries momentum because its internal energy flows are not symmetric. Here the asymmetry is created by the boost rather than by a field, and the consequence lands on position rather than on momentum.

How large the ambiguity is #

The reason nobody trips over this in ordinary mechanics is arithmetic rather than principle.

The Møller radius is an angular momentum divided by a mass and by the speed of light, and the speed of light in the denominator is large. A kilogram flywheel spinning at three thousand revolutions a minute has one of about $10^{-8}$ metres — real, and utterly beneath any measurement of where the flywheel is. The Earth’s is smaller still relative to its size.

For an elementary particle the numbers invert. An electron’s spin is $\hbar/2$ and its mass is tiny, so its Møller radius is $\hbar/2mc$, about $2\times10^{-13}$ metres. That is a hundred times the classical electron radius and vastly larger than any substructure experiments have failed to find. So the position of an electron is ambiguous at a scale far above anything else about it, and the ambiguity is not quantum in origin — this whole argument is classical relativity with a spin.

That has a real consequence in quantum mechanics. There is more than one candidate position operator for a relativistic particle with spin, they differ by about a Compton wavelength, and which one is “the” position depends on what is being asked. The classical ambiguity above is what the disagreement is made of.

Where the shift comes from, in one arrangement #

The disc is easier to believe with one concrete arrangement in front of it, and the simplest is not a spinning wheel but a pair of masses on a rod.

Take two equal masses at the ends of a light rod, moving in opposite directions perpendicular to it, so the pair has angular momentum and no net momentum. In their common rest frame the centre of energy is the middle of the rod, by symmetry.

Now watch from a frame moving along the rod. The mass whose motion now has a component along the boost has its speed changed differently from the other one — velocities do not simply add — so the two energies are no longer equal. The heavier-weighted end pulls the average towards it, and the centre of energy is no longer at the middle of the rod.

Reverse the boost and it shifts the other way. Boost perpendicular to the rod instead and it does not shift at all, which is why the displacement is a vector perpendicular to both the spin and the motion rather than a scalar. Every feature of the disc is present in this two-mass toy, and the Møller radius is the largest displacement the toy can produce.

Nothing internal has changed — the rod is the same rod, the masses are the same masses, and no force has acted. What changed is which events the observer calls simultaneous and how much energy each end carries, and the centre is an average over exactly those two things.

What is still true #

It would be easy to read all this as the centre of mass having been demolished, and it has not been. What survives is precise and is what the theorem is actually used for.

In any one frame the centre of energy is a perfectly good worldline, straight when the system is isolated, moving at $P/E$. Every calculation that uses the centre-of-mass frame — a collider’s energy budget, a decay’s kinematics, the reduced-mass trick — is done in one frame and is untouched.

The zero-momentum frame is unambiguous. Which frame has vanishing total momentum is a frame-independent question with one answer, and it is the frame in which the system’s mass is its energy. What is ambiguous is where the centre sits, not which frame is the rest frame.

And the ambiguity is bounded. The disc has a radius, and the radius is a definite property of the system. A statement about the centre’s location is meaningful to within $S/Mc$ and meaningless below it, which is a clean rule rather than a general vagueness.

The right summary is that a system’s position is a less robust idea than its momentum, its energy or its mass, and relativity is where that first becomes visible. It is also where nothing is allowed to be rigid comes from — an extended object with no unambiguous centre is not going to have an unambiguous shape either.

What the collider actually uses #

None of the above is an obstacle in practice, and it is worth saying exactly why, because the frame everybody computes in is the one this essay has been calling ambiguous.

A collider experiment works in the zero-momentum frame throughout. It never asks where the centre is; it asks what the total energy is in the frame where the momentum vanishes, which is the system’s invariant mass and is the same number for everybody. The collision that wastes most of the energy is that arithmetic and it involves no position at all.

The same is true of decay kinematics, of the reduced-mass trick, and of every scattering calculation: the useful content of “centre of mass” in relativistic practice is a velocity and not a place. Velocities transform cleanly, they are the same physical motion described differently in each frame, and the ambiguity in this essay never touches them.

Where positions do enter — in a beam optics calculation, or in locating a decay vertex — the work is done in the laboratory frame and stays there. A vertex is an event, and events are frame-independent; it is only the average of several events, weighted by frame-dependent numbers, that goes wrong. The lesson is not to avoid positions but to avoid averaging them, and almost every relativistic calculation is arranged that way without anyone having decided to arrange it.

Where the model stops #

The two-body figures are one-dimensional. Real collisions scatter into a plane at least, the energy shares depend on angle as well as speed, and the centre of energy moves in three dimensions. Nothing about the argument changes and every formula acquires vectors.

The collision is treated as instantaneous. During a real interaction the energy is partly in the field between the bodies, and the centre of energy accounts for that field energy too — which is what keeps the line straight during the collision as well as before and after. Drawing it as a point event hides the part of the bookkeeping that is doing the most work.

The Møller radius is a classical statement about a classical spinning body. Applying it to an electron uses the electron’s spin as though it were an internal circulation of energy, which is a picture the electron does not support. The conclusion about the position operator is right and the mechanism drawn is an analogy.

And gravity is absent. In general relativity there is no global notion of simultaneity to average over and no linear space of positions to average in, so the centre of energy has to be defined quasi-locally, and the several available definitions do not agree. The ambiguity described here is the flat-space shadow of a much larger one.

And a system has to be isolated for any of it to hold. The straight worldline is a consequence of conservation, so a body being pushed by anything outside itself has a centre of energy that accelerates, and a body exchanging radiation with its surroundings has one that is not even well defined until the radiation is included in the system. Deciding where a system ends is a larger part of the work than it appears, and the drawn collisions quietly assume the two bodies are the whole of it.

The disc is swept by inertial frames only. An accelerating observer’s notion of simultaneity is not one of the slices considered here, and it can put the centre outside the disc altogether. That is why the ambiguity is usually quoted with the qualification that it is over inertial frames, and why an accelerating spinning body needs a further condition imposed by hand — the choice that the clock that does not feel the turn shows accumulating consequences of its own.

What the pictures cannot show #

The collision figure draws worldlines and cannot draw the field between the bodies, which is where some of the energy is while they are interacting. The straight centre line is exact only because that energy is included, and the drawing shows the conclusion without the term that makes it true.

The displacement figure plots a distance against a frame speed as though a distance in one frame and a distance in another were commensurable. They are, here, because the displacement is perpendicular to the boost and perpendicular lengths are unchanged — but nothing in the picture says so, and the same plot drawn for a longitudinal quantity would be meaningless.

A third thing outside the pictures is the history of the argument. The disc was found by Møller in 1949, and the reason it took four decades after special relativity is that nobody had needed to ask where a spinning body was until the electron’s spin made the question unavoidable. The classical result and the quantum difficulty arrived within a few years of each other and from opposite directions, which is why the same length keeps appearing in both literatures under different names.

Where the ladder goes next #

The relativistic-dynamics ladder began with the push that does not point where the body goes, continued through the collision that wastes most of the energy, and reached nothing is allowed to be rigid. This rung asks where a system is and finds that the answer depends on who is asking. The rungs after it: the spin supplementary condition, which is the choice a relativistic spinning body’s equations require somebody to make; the Thomas precession, which is what that choice costs when the body accelerates; and the energy-momentum tensor, in which every quantity here is a component and the bookkeeping becomes automatic.

The habit worth carrying away is that some quantities are more robust than others. Energy, momentum and mass survive a change of frame in an orderly way and position does not, and a theory’s least robust quantity is usually the one carrying an assumption nobody stated.